Spectral convergence for a general class of random
matrices
Francisco Rubio, Xavier Mestre
To cite this version:
Francisco Rubio, Xavier Mestre.
Spectral convergence for a general class of random matrices.
STATISTICS & PROBABILITY LETTERS, 2011, 81 (5), pp.592.
<10.1016/j.spl.2011.01.004>. <hal-00725102>
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Accepted Manuscript
Spectral convergence for a general class of random matrices
Francisco Rubio, Xavier Mestre
PII:
DOI:
Reference:
S0167-7152(11)00011-3
10.1016/j.spl.2011.01.004
STAPRO 5879
To appear in:
Statistics and Probability Letters
Received date: 11 June 2009
Revised date: 27 July 2010
Accepted date: 8 January 2011
Please cite this article as: Rubio, F., Mestre, X., Spectral convergence for a general class of
random matrices. Statistics and Probability Letters (2011), doi:10.1016/j.spl.2011.01.004
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ACCEPTED MANUSCRIPT
CR
Francisco Rubio, Xavier Mestre
IPT
Spectral Convergence for a General Class of
Random Matrices
US
Centre Tecnològic de Telecomunicacions de Catalunya
Av. Carl Friedrich Gauss, 7, 08860 Castelldefels (Barcelona), Spain
Abstract
DM
AN
Let X be an M
N complex random matrix with i.i.d. entries having mean
zero and variance 1=N and consider the class of matrices of the type B = A +
R1=2 XTXH R1=2 , where A, R and T are Hermitian nonnegative de…nite matrices, such that R and T have bounded spectral norm with T being diagonal, and
R1=2 is the nonnegative de…nite square-root of R. Under some assumptions on the
moments of the entries of X, it is proved in this paper that, for any matrix
with
h bounded tracei norm and for each complex z outside the positive real line,
Tr
(B zIM ) 1
M (z) ! 0 almost surely as M; N ! 1 at the same rate,
where M (z) is deterministic and solely depends on ; A; R and T. The previous
result can be particularized to the study of the limiting behavior of the Stieltjes
transform as well as the eigenvectors of the random matrix model B. The study is
motivated by applications in the …eld of statistical signal processing.
EP
Introduction
TE
Key words: random matrix theory, Stieltjes transform, multivariate statistics,
sample covariance matrix, separable covariance model
AC
C
Consider an M N random matrix X such that the entries of N X are i.i.d.
standardized complex random variables with …nite 8 + " moment. Furthermore, consider an M M Hermitian nonnegative de…nite matrix R and its
nonnegative de…nite square-root R1=2 . Then, the matrix R1=2 XXH R1=2 can
be viewed as a sample covariance matrix constructed using the N columns
of the data matrix R1=2 X, namely having population covariance matrix R.
Moreover, consider also an N N diagonal matrix T with real nonnegative
entries. The matrix R1=2 XTXH R1=2 can be interpreted as a sample covariance matrix obtained by weighting the previous multivariate samples with the
entries of T.
Preprint submitted to Elsevier Science
26 July 2010
ACCEPTED MANUSCRIPT
M
1 X
I(
M m=1
),
m (B)
(1)
CR
FBM ( ) =
IPT
In this paper, we are interested in the asymptotic behaviour of certain spectral
functions of the random matrix model B = A + R1=2 XTXH R1=2 , where A is
an N N Hermitian nonnegative de…nite matrix, as the dimensions M and
N grow large at the same rate. Consider the empirical distribution function
of the eigenvalues of B, i.e.,
M
1 X
dFBM
=
z
M m=1
R
1
m (B)
=
h
1
Tr (B
M
DM
AN
mF (z) =
Z
US
where m (B) stands for the mth eigenvalue of B and 1 (B) : : :
M (B).
From the connection between vague convergence of distributions and pointwise convergence of Stieltjes transforms (see, e.g., [3] and [7, Proposition 2.2]),
almost sure convergence of the (random) distribution function FBM can be established by showing convergence of the associated Stieltjes transform, de…ned
for each z 2 C+ = fz 2 C : Im fzg > 0g as
z
zIM )
1
i
.
(2)
EP
TE
Under the condition that M=N has a …nite non-zero limit, convergence with
probability one of the empirical spectral distribution (ESD), i.e., the empirical distribution of the eigenvalues, of some special cases of B towards a limit
nonrandom distribution has been established in the random matrix theory literature under the assumption of the matrices R; T and A having, in each case,
an ESD which converges almost surely to a probability distribution (possibly
defective in the case of A). More speci…cally, using the fact that the limit
of FBM is uniquely characterized by that of mF (z), vague convergence in the
cases of A = 0M M and T = IN ; R = IM ; and A = 0M M is provided
in
n
o
M
[13], [14], and [12,9], respectively, by proving tightness of the sequence FB
(the convergence is weakly to a proper probability measure if A = 0M M ).
The reader is also referred to [5] for similar results and other random matrix
models.
AC
C
In order to extend the previous spectral convergence results to the asymptotic
behavior of the eigensubspaces of B, we can consider the following empirical
distribution function, namely,
M
HB
( )=
M
X
m=1
jam j2 I(
m (B)
),
(3)
where am is the mth entry of the vector a = UH , with U 2 CM M being the
matrix of orthonormal
B, and an arbitrary nonrandom vector
n eigenvectors of o
N
M
in the unit sphere
2 C : k k = 1 . Clearly, HB
is a random probability
2
ACCEPTED MANUSCRIPT
mH (z) =
Z
M
dHB
( )
=
z
R
H
(B
zIM )
1
IPT
distribution function with Stieltjes transform given by
.
(4)
CR
In particular, spectral functions of the form of (3) or, equivalently, (4) were
considered in [1] (see also references therein) to study the limiting eigenvectors
of sample covariance matrices. Note further that, if am = M1 , m = 1; : : : ; M ,
M
then HB
and FBM coincide.
1
DM
AN
US
Motivated by practical applications, for instance in the …elds of statistical signal processing, wireless communications and quantitative …nance, where estimates of certain functions of the eigenvalues and eigenvectors of the random
matrix model B are very often of interest, in this paper the convergence of (2)
and (4) is extended to the convergence of more general spectral functions of
B. Moreover, for the purpose of practical applicability in realistic settings, the
assumption of R; T and A having convergent ESD is dropped in the following.
Main result
The remainder of the paper is devoted to the proof of the following theorem.
Theorem 1 Assume the following:
AC
C
EP
TE
p
(a) X is an M N random matrix such that the entries of N X are i.i.d.
complex random variables with mean zero, variance one, and …nite 8 + "
moment, for some " > 0.
(c) A and R are M M Hermitian nonnegative de…nite matrices, with the
spectral norm (denoted by k k) of R being bounded uniformly in M , and T is
an N N diagonal matrix with real nonnegative entries uniformly bounded
in N .
(d) B = A + R1=2 XTXH R1=2 , where R1=2 is the nonnegative de…nite squareroot of R.
(e)
is an arbitrary nonrandom M M matrix, whose trace norm (i.e.,
Tr
1=2
H
k ktr ) is bounded uniformly in M .
Then, with probability one, for each z 2 C R+ , as N = N (M ) ! 1 such
that 0 < lim inf cM < lim sup cM < 1, with cM = M=N ,
Tr
h
(B
zIM )
1
(A + xM (eM ) R
3
zIM )
1
i
! 0,
(5)
ACCEPTED MANUSCRIPT
xM (eM ) =
h
i
1
Tr T (IN + cM eM T) 1 ,
N
IPT
where xM (eM ) is de…ned as
(6)
CR
and eM = eM (z) is the Stieltjes transform of a certain positive measure on
R+ with total mass M1 Tr [R], obtained as the unique solution in C+ of the
equation
h
i
1
eM =
Tr R (A + xM (eM ) R zIM ) 1 .
(7)
M
US
Corollary 1 (Limiting Stieltjes transform of ESD’s) Let
= M1 IM . Then,
Theorem 1 states the asymptotic convergence of the Stieltjes transform of the
ESD of B de…ned in (2).
DM
AN
Remark 1 (Sample covariance matrices) If A = 0M M and T = IN , then
Corollary 1 is equivalent to the convergence result for the Stieltjes transform
of the ESD of sample covariance matrices provided in [16,4][13, Theorem 1.1].
Remark 2 (Marchenko-Pastur law) If R = IM , then the result in Corollary
1 reduces to that obtained in [10][14, Theorem 1.1] under more general conditions.
Remark 3 (Separable covariance model) If A = 0M M , then Corollary 1
coincides with the result in [12, Theorem 1] (see also [9, Theorem 2]), which
represents the special case of [7, Theorem 2.5] corresponding to a separable
variance pro…le.
TE
H
Corollary 2 (Asymptotic convergence of eigenvectors) Let
=
, with
M
2 C being an arbitrary nonrandom unit vector. Then, Theorem 1 establishes the convergence for the class of Stieltjes transforms de…ned in (4).
EP
Remark 4 The result in Corollary 2 for the cases R = IM ; T = IN and
A = 0M M ; T = IN has been previously reported in [11, Theorem 1 and
Theorem 5], and [1, Theorem 1] under more generic assumptions.
2
AC
C
As a …nal remark, we notice that Theorem 1 holds verbatim if the matrices
A, R, T, X and
have real-valued entries.
Applications to Array Signal Processing
The random matrix model introduced in previous sections is of fundamental
interest in applications involving spatio-temporal statistics as well as separable
covariance models. Consider for instance a collection of N narrowband signal
observations that are obtained by sampling across a linear antenna array with
4
ACCEPTED MANUSCRIPT
US
CR
IPT
M sensors, and which can be modelled as yn = sn s + nn , n = 1; : : : ; N , where
sn 2 C is the signal waveform, s 2 CM is the associated signature vector,
and nn 2 CM models the contribution of the interference and noise. Conventionally, sn and nn are assumed to be Gaussian, temporarily uncorrelated and
mutually independent, with variance and covariance matrix given respectively
by s2 and Rn . Under these assumptions, the observations can be modelled as
yn = R1=2 n , R = s2 ssH + Rn is the covariance matrix of the array samples,
and the vectors n 2 CM consist of i.i.d. standardized (i.e., with mean zero
and variance one) Gaussian random entries. In the array processing literature, the minimum variance distortionless response (MVDR) spatial …lter or
1
beamformer is de…ned in terms of R as wMVDR = sH R 1 s
R 1 s (see, e.g.,
[15, Chapter 6]). In practice, the true covariance matrix R is unknown, and
^ must be used instead
therefore a sample estimate, henceforth denoted by R,
for implementation purposes. A relevant measure of the …lter performance is
the output signal-to-interference-plus-noise ratio (SINR), which for a given
beamformer w
^ is de…ned as
!
1
DM
AN
SINR (w)
^ =
w
^ H Rw
^
2
2 jw
H
s ^ sj
1
.
We note that SINR (w)
^ is invariant under scaling and consider the …lter w
^ =
1
^
R s, namely de…ned in terms of a generic covariance sample estimator given
^ = YWYH + R0 , where Y = y1
by R
yN 2 CM N , W is an N N
3
AC
C
EP
TE
diagonal nonnegative de…nite matrix, R0 is an M M Hermitian nonnegative
matrix, and ; 2 R. In particular, if W = IN , = 1 and R0 = 0M M ,
^ is the sample covariance matrix or maximum likelihood estimator
then R
of R; moreover, if W is a data windowing diagonal matrix with nth entry
given by wn = exp (N n), = 1 and R0 = IM , then w
^ implements the
recursive least-squares beamformer with exponential weighting and diagonal
^ has the form
loading factor (see, e.g., [15, Chapter 7]); …nally, if W = IN , R
of a James-Stein shrinkage covariance estimator, where R0 is the shrinkage
target or prior information about R and ; are the shrinkage intensity parameters. Interestingly enough, a uni…ed approach to the analysis of the SINR
performance of the previous MVDR sample beamformers can be provided by
applying Theorem 1 in order to …nd a deterministic asymptotic equivalent of
SINR (w).
^
Preliminary results
In this section, we introduce some preparatory lemmas and auxiliary technical
results that will be needed in the proof of Theorem 1. The …rst one has a
5
ACCEPTED MANUSCRIPT
N
o
denote a collection of (possibly dependent)
h
pi
max E yn(N )
1 n N
Kp
,
N 1+
CR
n
Lemma 1 Let yn(N ) ; 1 n
random variables such that
IPT
straightforward proof based on Markov’s and Minkowski’s inequalities (see,
e.g., proof of Lemma 4.1 in [6] for details).
for some constants p
1, > 0 and Kp (in the sequel, constants Kp may
take di¤erent values at each appearance) depending on p but not on N . Then,
almost surely as N ! 1,
US
N
1 X
y (N ) ! 0.
N n=1 n
E
h
H
C
Tr [C]
pi
DM
AN
Lemma 2 ([2, Lemma 2.7]) Let 2 CM denote a random vector with i.i.d.
entries having mean zero and variance one, and C 2 CM M an arbitrary
nonrandom matrix. Then, for any p 2,
h
i
h
E j j4 Tr CCH
Kp
i
p=2
h
i
+ E j j2p Tr
p=2
CCH
where denotes a particular entry of and the constant Kp does not depend
on M , the entries of C, nor the distribution of .
and C be de…ned as in Lemma 2. Then, for any p
E
h
H
C
pi
h
kCkptr K1;p + K2;p E j j2p
TE
Lemma 3 Let
i
2,
.
EP
PROOF. Let and be two complex random variables with …nite pth-order
moment. Using Jensen’s inequality, note that
E [j + jp ]
2p
1
fE [j jp ] + E [j jp ]g .
(8)
Furthermore, consider the following two useful inequalities:
CCH
p=2
AC
C
Tr
h
Tr CCH
i
p=2
Tr
CCH
1=2
p
= kCkptr ,
(9)
and, for B1 and B2 two arbitrary square complex matrices [8, Chapter 3]
jTr [B1 B2 ]j
kB1 B2 ktr
kB1 ktr kB2 k ,
(10)
which will be used repeatedly in the sequel. Now, using (8) we write
E
h
H
C
pi
2p
1
n h
E
H
C
6
Tr [C]
pi
o
+ jTr [C]jp .
(11)
,
ACCEPTED MANUSCRIPT
h
Moreover, using the …rst inequality in (9) and Jensen’s inequality as Ep=2 j j4
h
i
E
h
H
Tr [C]
C
pi
h
Kp Tr CCH
i
p=2
IPT
E j j2p , from Lemma 2 we get
i
h
i
E j j2p .
(12)
n
CR
Finally, the result of the lemma follows by applying to (11) the inequality (12)
along with the …rst inequality in (10) with B1 = C and B2 = IM .
o
US
Lemma 4 Let U = n 2 CM ; 1 n N denote a collection of i.i.d. random vectors de…ned as in Lemma 2, and whose entries are assumed to have
…nite 4n+ " moment, " > 0. Furthermore,
consider a collection of random mao
M M
trices C(n) 2 C
; 1 n N such that, for each n, C(n) may depend on
all the elements of U except for n , and C(n)
M . Then, almost surely as N ! 1,
h
i
DM
AN
N
1 X
N n=1
is uniformly bounded for all
tr
H
n C(n) n
Tr C(n) ! 0.
h
(13)
i
PROOF. Write n = nH C(n) n Tr C(n) and let Fn denote the -…eld generated by the random vectors f 1 ; 2 ; : : : ; n g. Now, notice that f n ; 1 n N g
forms a martingale
n N g.
h di¤erencei array with respect to the …ltration fFn ; 1
Indeed, since E n nH jFn 1 = IM , observe that
n
jFn
1]
=E
h
H
n C(n) n
jFn
TE
E[
1
i
h
h
i
E Tr C(n) jFn
1
i
= 0.
Consequently, Burkholder’s inequality (see, e.g., [2, Lemma 2.1]) implies
E4
N
X
p
n
n=1
8 2
N
<
h
X
4
Kp E
E j
:
3
5
EP
2
2
nj
n=1
jFn
1
i
!p=2 3
" N
X
5+E
j
p
nj
n=1
#9
=
;
,
AC
C
for any p 2. Then, since n and C(n) are independent for each n, using (12)
and the properties of conditional expectation, it is easy to check that
h
E j n j2 jFn
1
i
=E
h
H
n C(n) n
Tr C(n)
i
2
jFn
h
h
where we have used the fact that the random variable Tr C(n) CH
(n)
is bounded uniformly in M by assumption. Similarly, we also get
E [j n jp ] = E
h
H
n C(n) n
h
Tr C(n)
7
i pi
i
Kp E j j4 ,
1
h
i
Kp E j j2p .
i
C(n)
2
tr
ACCEPTED MANUSCRIPT
Hence, we can …nally write
N
1 X
4
E
N n=1
p
n
3
5
h
i
1
4
K
N
E
j
j
1;p
Np
p=2
h
+ K2;p N E j j2p
i
h
i
K1;p
K2;p
+ p 1 ,
p=2
N
N
IPT
2
E j j2p
and the result follows from the Borel-Cantelli lemma with p = 2 + "=2, " > 0.
zIM )
1
=
B2 + qqH
1
zIM
qH (B2 zIM ) 1 B1 (B2 zIM )
1 + qH (B2 zIM ) 1 q
1
US
Tr B1 (B2
CR
Lemma 5 ([2, Lemma 2.7]) Consider two N N matrices B1 and B2 , with
B2 being Hermitian, and 2 R, q 2 CN . Then, for each z 2 C+ ,
q
kB1 k
.
Im fzg
DM
AN
Lemma 6 Let m (z) be the Stieltjes transform of a certain measure on R+ ,
and 2 R+ . Then, for each z 2 C+ ,
1 + m (z)
jzj
.
Im fzg
PROOF. The bound follows from the fact that, for any 2 R+ , z(1+ m(z))
is the Stieltjes transform of a measure on R+ with total mass [12], whose
absolute value is therefore bounded by = Im fzg [7, Proposition 2.2].
Proof of Theorem 1
TE
4
EP
In this section, we give a proof of Theorem 1. Let
1 1=2
R
T
N
H
R1=2 =
N
1 X
tn R1=2
N n=1
=
H 1=2
n n R
p
N X and write
N
1 X
yn ynH ,
N n=1
AC
C
where tn is the nth diagonal entry of T, and n 2 CM is the nth column
vectorh of . Moreover, let
B(n) = B N1 yn ynH , and consider the map xM (e) =
i
1
Tr T (IN + cM eT) 1 : C+ ! C = fz 2 C : Im fzg < 0g. Indeed, note
N
that
N
1 X
cM t2n
.
(14)
Im fxM (e)g = Im feg
N n=1 j1 + tn cM ej2
For the sake of notational convenience, we will use the following de…nitions:
Q (z) = (B
zIM )
1
, Q(n) (z) = B(n)
8
zIM
1
, P (e) = (A + x (e) R
zIM )
1
.
ACCEPTED MANUSCRIPT
1=2
1=2
1
i IM
1
1
.
Im fzg
1
(15)
CR
kP (e)k =
IPT
It is easy to see that both kQ (z)k and Q(n) (z) are upper-bounded by
1= Im fzg. Equivalently, de…ne
= A + Re fx (e)g R Re fzg IM and
=
Im fzg IM Im fx (e)g R, which are, respectively, Hermitian and Hermitian
positive de…nite, and note that
Now, consider the equality
1
(z) = B zIM = A+xM (^
eM ) R zIM +
1 1=2
R
T
N
DM
AN
Q
US
Furthermore, let e^M (z) = M1 Tr [RQ (z)], and notice (see, e.g., [7,12]) that
e^M = e^M (z) is the Stieltjes transform of a certain measure on R+ with total mass M 1 Tr [R]
kRksup , where with the subscript sup we denote the
supremum of the sequence, i.e., here, kRksup = supM 1 kRk (in the sequel, we
will similarly use inf for the in…mum).
H
R1=2
xM (^
eM ) R.
We proceed by factoring the di¤erence of inverses as
P (^
eM )
1 1=2
R
T
N
Q (z) = P (^
eM )
H
R1=2
xM (^
eM ) R Q (z) ,
(16)
where we have used the resolvent identity, i.e., B1 B2 = B1 B2 1 B1 1 B2 .
Furthermore, the middle factor on the RHS of (16) can be expanded as
R1=2
xM (^
eM ) R Q (z) =
TE
H
=
EP
1 1=2
R
T
N
=
N
1 X
tn R1=2
N n=1
N
1 X
tn R1=2
N n=1
H 1=2
Q (z)
n n R
xM (^
eM ) RQ (z)
H 1=2
Q (z)
n n R
tn RQ (z)
1 + tn cM e^M (z)
N
tn R1=2 n nH R1=2 Q(n) (z)
1 X
N n=1 1 + tn N1 nH R1=2 Q(n) (z) R1=2
n
tn RQ (z)
,
1 + tn cM e^M (z)
AC
C
where, in the last equality, we have used the Sherman-Morrison-Woodbury
identity for rank augmenting matrices in order to write
Q(z) = Q(n) (z)
tn N1 Q(n) (z)R1=2 n nH R1=2 Q(n) (z)
,
1 + tn N1 nH R1=2 Q(n) (z) R1=2 n
(17)
along with the following useful inequality (see [14, Eq. (2.2)])
H 1=2
n R
Q(n)1 (z) + tn N1 R1=2
1
H 1=2
n n R
9
=
1
1+
tn N1 nH R1=2 Q(n)
(z) R1=2
n
H 1=2
Q(n) (z):
n R
ACCEPTED MANUSCRIPT
1 1=2
R
T
N
H
R1=2
xM (^
eM ) R Q (z) =
N
1 X
tn
=
R Q(n) (z)
N n=1 1 + tn cM e^M (z)
Q (z)
1 H 1=2
R Q(n) (z) R1=2 n
N n
tn N1 nH R1=2 Q(n) (z) R1=2 n
Tr [RQ (z)]
1+
N
tn
1 X
R1=2
N n=1 1 + tn cM e^M (z)
CR
1
N
N
tn
1 X
tn
+
N n=1 1 + tn cM e^M (z)
+
IPT
Accordingly, observe that we can further write
H 1=2
Q(n)
n n R
1
M
R1=2
H 1=2
Q(n)
n n R
(z)
RQ(n) (z) .
(z)
(e) =
Tr [
and
DM
AN
M
US
Tr [RP (e)] be a function
mapping C+ into )C+ and fM (e) =
(
jzj
+
e
K , for some …nite,
M (e). Moreover, we de…ne D = z 2 C :
Im fzg
positive K large enough. Next, we prove that, almost surely as M; N ! 1
with 0 < lim inf cM < lim sup cM < 1, for each z 2 D,
Let
P (^
eM ))] ! 0,
(Q (z)
(18)
fM (^
eM ) ! 0.
(19)
To that e¤ect, note that it is enough to show the following almost sure convergence to zero on D of the following quantities:
TE
N
h
tn
1 X
Tr ~ R Q(n) (z)
N n=1 1 + tn cM e^M (z)
N
1 X
tn
1
Tr [RQ (z)]
N n=1 1 + tn cM e^M (z) N
EP
N
h
1 X
tn
Tr ~ R1=2
N n=1 1 + tn cM e^M (z)
1
N
Q (z)
H 1=2
Q(n)
n R
H 1=2
Q(n)
n n R
(z) R
(z)
i
1=2
,
(20)
1=2
n
RQ(n) (z)
tn nH R1=2 Q(n) (z) ~ R
n
1 + tn N1 nH R1=2 Q(n) (z) R1=2
(21)
i
,
(22)
AC
C
where ~ = P (^
eM ), with being an M M matrix with uniformly bounded
trace norm, which can either be = or = M1 R (note that kM 1 Rktr =
M 1 Tr [R] kRksup ). Before proceeding, we note that Lemma 6 yields
max
1 n N
tn
1 + tn cM e^M (z)
jzj kTksup
.
Im fzg
(23)
Consider the convergence of (20). First, notice that, for each n, N1 nH R1=2 Q(n) (z) R1=2
can be viewed as the Stieltjes transform of a certain measure on R+ (see, e.g.,
10
n
,
n
ACCEPTED MANUSCRIPT
[12]). Hence, from Lemma 6 we have
1 n N
1+
tn
tn N1 nH R1=2 Q(n)
(z) R1=2
jzj kTksup
.
Im fzg
IPT
max
n
(24)
1 n N
for any M
h
H 1=2
Q(n)
n R
(z) ~ Z(n) R1=2
M matrix Z(n) independent of
pi
n
Kp ,
US
max E
CR
Now, applying (17) to expand the term Q(n) (z) Q (z) and using (23) and
(24), the result follows readily by Lemma 1 with p 2 together with the fact
that, for each z 2 D,
p
such that Z(n)
n
(25)
Kp ,
sup
(n)
e^M
in particular for Z(n) = RQ(n) (z). Let us now prove (25). Let
(z) =
h
i
1
Tr RQ(n) (z) , which is a Stieltjes transform with same properties as e^M (z),
M
(n)
(n)
P (^
eM ) = P e^M +
(n)
e^M ; e^M
M
where we have de…ned
M
(e1 ; e2 ) =
h
cM
Tr R Q (z)
M
h
1
Tr T2 (IN + cM e1 T)
N
1
Q(n) (z)
i
that
(n)
P (^
eM ) RP e^M
(26)
1
i
,
jzj kTksup
Im fzg
tn
tn
max
(n)
1 n N 1+t c e
1 + tn cM e^M (z)
n M ^M (z)
(n)
e^M ; e^M
P e^M
(IN + cM e2 T)
TE
and note from Lemma 6 that
M
DM
AN
and notice by applying twice the resolvent identity to P (^
eM )
!2
.
H 1=2
Q(n)
n R
AC
C
h
EP
(27)
Now, we write ~ = P (^
eM ) using the two terms on the RHS of (26). From
(n)
the …rst term, by Lemma 3 with C = R1=2 Q(n) (z) P e^M Z(n) R1=2 , we
obtain
max E
1 n N
(z)
P
(n)
e^M
Z(n) R
1=2
n
pi
Regarding the second term in (26), i.e., P (^
eM )
Lemma 5 we have
P (^
eM )
P
(n)
e^M
Z(n) Q(n) (z)
p
11
Kp kRkpsup k kptr;sup
(Im fzg)2p
P
(n)
e^M
.
(28)
, from (27) and
Kp jzj kRksup kTksup
N p (Im fzg)6p
2p
.
(29)
,
ACCEPTED MANUSCRIPT
Hence, from the second term we get
(a)
H 1=2
Q(n)
n R
(z)
Kp jzj kRksup kTksup
(Im fzg)6p
(b)
Kp kRk3p
sup jzj kTksup
2p
(n)
P (^
eM )
P e^M
Z(n) R1=2
k ktr;sup
max E
1
N
1 n N
2p
(Im fzg)6p
k kptr;sup
pi
IPT
1 n N
h
n
p
H
n R n
CR
max E
,
(30)
US
where (a) in (30) follows by the Cauchy-Schwarz inequality along with (29) and
the second inequality in (10), and (b) follows by Lemma 3 with C = N 1 R.
Finally, (25) follows by applying (8) together with (28) and (30).
2
1
E4
Tr [RQ (z)]
N
E1=2
"
1
N
1
Tr [RQ (z)]
N
DM
AN
We now prove the convergence of (21). Using the Cauchy-Schwartz inequality,
we write
H 1=2
Q(n)
n R
1
N
(z) R
1=2
1=2
tn nH R1=2 Q(n) (z) ~ R
n
1 H 1=2
1=2
1 + tn N n R Q(n) (z) R
n
2p
H 1=2
Q(n)
n R
(z) R1=2
n
#
2
E1=2 6
4
1
p3
5
n
TE
Now, notice that from (24) and (25) with Z(n) = IM the second factor of the
RHS of the previous inequality is uniformly bounded on D for any p
1.
Furthermore, regarding the …rst factor, we rewrite Q (z) by using the two
terms on the RHS of (17) and get, on the one hand (cf. Lemma 5),
2
EP
1 tn N1 nH R1=2 Q(n) (z) RQn (z)R1=2
max E 4
1 n N
N 1 + tn N1 nH R1=2 Q(n) (z) R1=2 n
and on the other hand, using (12) with C = N
h
i
1
Tr RQ(n) (z)
N
AC
C
"
max E
1 n N
1
N
2p
n
3
1
N 2p
5
1=2
R
1=2
(z) R
Finally, the result follows from Lemma 1 with p
with (31) and (32).
1=2
kRksup
Im fzg
Q(n) (z) R
2p
H 1=2
Q(n)
n R
#
1=2
!2p
,
(31)
we get
Kp kRksup
n
Im fzg
(32)
2 by applying (8) along
1
Np
In order to prove the convergence of (22), by the triangular inequality, it is
enough to show the almost sure convergence to zero on D of the following
12
3
2p
1=2
tn nH R1=2 Q(n) (z) ~ R
n
7
5.
1 H 1=2
1=2
+ tn N n R Q(n) (z) R
n
!2p
.
ACCEPTED MANUSCRIPT
quantities:
n=1
tn
(n)
1 + tn cM e^M (z)
(n)
1 + tn cM e^M (z)
!
H 1=2
Q(n)
n R
h
IPT
1
N
N
X
tn
1=2
(z) ~ R
Tr RQ(n) (z) ~
n
(33)
H 1=2
Q(n)
n R
(z)
(n)
R1=2
P e^M
n
h
Tr RQ(n) (z)
CR
N
1 X
tn
N n=1 1 + tn cM e^M (z)
n
US
N
tn
1 X
(n)
H 1=2
P (^
eM ) P e^M
R1=2
Q(n) (z)
n R
(n)
N n=1 1 + tn cM e^M (z)
N
h
i
tn
1 X
(n)
Tr RQ(n) (z)
P e^M
P (^
eM ) .
(n)
N n=1 1 + tn cM e^M (z)
(n)
P e^M
(34)
i
i
,
,
, (35)
(36)
Regarding (33), using (27) along with Lemma 5, note …rst that
i
DM
AN
h
2
tn
tn
1 cM tn Tr R Q(n) (z) Q (z)
=
1 + tn cM e^M (z) 1 + tn cM e^(n)
N (1 + tn cM e^M (z)) 1 + tn cM e^(n)
M (z)
M (z)
kRksup jzj kTksup
N (Im fzg)3
(37)
Convergence of (33) follows by Lemmah 1 with p 2i by applying (37) and (8)
along with (25) with Z(n) = IM and Tr RQ(n) (z) ~
kRk jzj k ktr;sup = (Im fzg)2
(cf. inequality (10)). On the other hand, convergence of (34) follows directly
(n)
1
(n)
Tr RQ(n) (z)
P
(n)
e^M
P (^
eM )
EP
h
TE
by Lemma 4 with C(n) = tn 1 + tn cM e^M (z)
R1=2 Q(n) (z) P e^M R1=2 .
Convergence of (35) follows from Lemma 1 with p 2 by applying (26) and
(n)
using (23) and (27) along with Lemma 5, and (25) with Z(n) = RP e^M .
Finally, (36) vanishes almost surely by Lemma 1 with p 2 using (23) along
with (cf. inequality (10))
i
p
cpM kRk3p
sup jzj kTksup
2p
N p (Im fzg)6p
k kptr;sup
,
where we have used (29) with Z(n) = R. This proves (18) and (19).
AC
C
We next show that P (^
eM ) in (18) and (19) can be replaced by P (eM ), where
eM is the unique deterministic equivalent in the statement of Theorem 1. From
(19) and fM (eM ) = 0 we clearly have that fM (^
eM ) fM (eM ) ! 0, for each
z 2 D, and so we only need to show that this implies e^M eM ! 0. Indeed,
notice that fM (^
eM ) fM (eM ) = (^
eM eM ) (1
eM ; eM )), where we have
M (^
de…ned
1
Tr [RP (e1 ) RP (e2 )] .
(38)
M (e1 ; e2 ) = M (e1 ; e2 )
M
Observe that M (eM ; eM ) = 1. Moreover, after some algebraic manipulations
13
2
.
ACCEPTED MANUSCRIPT
we get fM (^
eM )
eM )2
fM (eM ) = (^
eM
M
(^
eM ; eM ), where
IPT
N
1 X
c2M t3n
1
Tr [RP (eM ) RP (^
eM )]
2
N n=1 (1 + tn cM eM ) (1 + tn cM e^M ) M
N
2
h
i
1 X
cM tn
1
eM ; eM )
Tr (RP (eM ))2 RP (^
eM ) ,
M (^
N n=1 1 + tn cM eM M
CR
eM ; eM ) =
M (^
such that, using the fact that eM is the Stieltjes transform of a certain measure
on R+ (cf. Section 4.2), by the triangular inequality and the application of
Lemma 6 as in (23) along with inequality (10),
M
(^
eM ; eM )j
2
jzj kTksup
(Im fzg)5
3
jzj kRksup kTksup
US
j
cM kRksup
1+
(Im fzg)2
!
.
DM
AN
This implies that e^M eM ! 0 almost surely for each z of a countable family
with an accumulation point in a compact subset of D. Now, as a consequence
of being Stieltjes transforms of bounded measures on R+ , we have that both
e^M (z) and eM (z) are analytic on C R+ (see [7, Proposition 2.2]), and so is
2
hM (z) = e^M (z) eM (z). Moreover, we have that jhM (z)j dist(z;R
+ ) , where
dist stands for the Euclidean distance (see again [7, Proposition 2.2]). Thus,
fhM g is a normal family and by Montel’s theorem there exists a subsequence
which converges uniformly on each compact subset of C R+ to an analytic
function which, from above, vanishes almost surely on C R+ . Thus, the entire
sequence converges uniformly to zero on each compact subset of C R+ , and
so jhM (z)j ! 0 for each z 2 C R+ .
gM (eM ) = (eM
e^M )
M
gM (eM ) ! 0 for each
(^
eM ; eM ) Tr [ P (^
eM ) RP (eM )] ,
EP
gM (^
eM )
TE
De…ne gM (e) = Tr [ P (e)]. The fact that gM (^
eM )
+
z 2 C R follows …nally by observing that
AC
C
and noting from (27) and jTr [ P (^
eM ) RP (eM )]j kRksup k ktr;sup = (Im fzg)2
(cf. inequality (10)) that the factor multiplying eM e^M in the RHS is bounded
in absolute value uniformly in M .
4.1 Uniqueness of the limit
In order to prove uniqueness in C+ of the solution of equation (7), assume
that we have two solutions e1 ; e2 2 C+ and, using the resolvent identity, write
e1 e2 =
(xM (e1 )
xM (e2 ))
1
Tr [RP (e1 ) RP (e2 )] = (e1
M
14
e2 )
M
(e1 ; e2 ) .
ACCEPTED MANUSCRIPT
'M (ei ) =
IPT
Assume e1 6= e2 , so that, necessarily, M = M (e1 ; e2 ) = 1. Using the CauchySchwarz inequality, we have j M j2 'M (e1 ) 'M (e2 ), where
N
h
i
1 X
cM t2n
1
H
Tr
P
(e
)
RP
(e
)
R
, i = 1; 2.
i
i
N n=1 j1 + tn cM ei j2 M
Then, from (14) we note that
N
1 X
cM t2n
=
N n=1 j1 + tn cM ei j2
CR
Im fx (ei )g
.
Im fei g
(39)
(40)
On the other hand, using the fact that x (e) 2 C for each z 2 C+ along with
we can conclude that
Im fei g
.
Im fx (ei )g
(41)
DM
AN
h
i
1
Tr PH (e1 ) RP (e1 ) R <
M
US
h
i 1
h
i
Im fei g
Im fzg
1
=
Tr PH (e1 ) RP (e1 )
Tr PH (e1 ) RP (e1 ) R ,
Im fxM (ei )g
Im fxM (ei )g M
M
Hence, it is clear from (39) by using (40) and (41) that 'M (ei ) < 1, i =
1; 2, and so M < 1, contradicting the fact that M = 1. Therefore we must
necessarily have e1 = e2 .
4.2 Existence of a deterministic asymptotic equivalent
8
<
TE
We use the …xed point theorem to show that the equation e = M (e) presents
at least one solution in C+ . Let M = max1 m M j m (A) zj, which is strictly
positive by assumption for all M and each z 2 C R+ . Consider the open set:
2
v
39
EP
u
=
kRksup
u
4c
41 + t1 + inf inf 5 .
= e 2 C+ : Im feg >
:
2cinf inf
kRksup ;
AC
C
We show that, for any two values e1 ; e2 2 , j M (e1 )
e2 j,
M (e2 )j < je1
so that the mapping M is contractive when constrained to . The …xed point
theorem guarantees that M presents a …xed point in , which guarantees the
existence of a solution of the equation e = M (e) in
C+ .
Observe …rst that we can write, using the resolvent identity, M (e1 ) M (e2 ) =
(e1 e2 ) M (e1 ; e2 ). Hence, it is su¢ cient to see that j M (e1 ; e2 )j < 1, or alternatively, from the Cauchy-Schwarz inequality, that 'M (e) < 1 for any e 2 .
Note that we can write
h
i
1
Tr PH (e) RP (e) R
M
R1=2 P (e) R1=2 = kRk kA + x (e) R
15
zIM k
1
.
(42)
ACCEPTED MANUSCRIPT
jx (e)j
kB1 + B2 k
kB1 k + kB2 k, together with
N
1 X
tn
N n=1 j1 + tn cM ej
and the fact that kA
zIM k
inf ,
kA + x (e) R
N
1 X
tn
N n=1 tn cM Im feg
we have
1
,
cinf Im feg
1
zIM k
1
,
cinf Im feg
(43)
IPT
jIm fxgj
kB2 k
CR
Now, using kB1 k
inf
(44)
jIm fx (e)gj
Im feg
kRksup
US
which is clearly positive on by assumption. Now, using (14) and the above
inequalities, in order to show that 'M (e) < 1, it is su¢ cient to prove that
1
cinf Imfeg
inf
< 1,
DM
AN
or, equivalently, cinf inf Im2 feg Im feg kRksup kRksup > 0, which is always
the case for each e 2 , and this concludes the proof of the existence of eM .
Finally, we show that eM (z) is the Stieltjes transform of a certain bounded
measure on R+ with total mass M1 Tr [R]. Indeed, notice that eM (z) is analytic
on C+ and, additionally,
Im feM g =
and
h
1
i
'M (eM )
i
,
!
h
i
tn
1
H
Tr P (eM ) RP (eM ) A +
Tr
P
(e
)
RP
(e
)
R
M
M
2M
n=1 j1 + tn cM eM j
.
1 'M (eM )
H
TE
Im fzeM g =
Im fzg
1
M
h
Im fzg M1 Tr PH (eM ) RP (eM )
1
N
N
P
AC
C
EP
Now, since 'M (eM ) < 1 on C+ (cf. Section 4.1) and since A is assumed to
be nonnegative de…nite, we conclude that both eM (z) and zeM (z) map C+
into C+ , and the claim follows from [7, Proposition 2.2] using the fact that
(y = Im fzg) limy!+1 i yeM (i y) = M1 Tr [R].
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ACCEPTED MANUSCRIPT
IPT
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US
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AC
C
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